This paper is part of a larger project about the relation between mathematics and transcendental philosophy that I think is the most interesting feature of Kant’s philosophy of mathematics. This general view is that in the course of arguing independently of mathematical considerations for conditions of experience, Kant also establishes conditions of the possibility of mathematics. My broad aim in this paper is to clarify the sense in which this is an accurate description of Kant’s view of the relation between (...) mathematics and transcendental philosophy. (shrink)

In her recent book, Realism in mathematics, Penelope Maddy attempts to reconcile a naturalistic epistemology with realism about set theory. The key to this reconciliation is an analogy between mathematics and the physical sciences based on the claim that we perceive the objects of set theory. In this paper I try to show that neither this claim nor the analogy can be sustained. But even if the claim that we perceive some sets is granted, I argue that Maddy's account fails (...) to explain the key issue faced by an epistemology for mathematics, namely the step from knowledge of the finite to knowledge of the infinite. (shrink)

In this paper I argue that Kant's distinction in the Inaugural Dissertation between the sensible and the intelligible arises in part out of certain open questions left open by his comparison between mathematics and metaphysics in the Prize Essay. This distinction provides a philosophical justification for his distinction between the respective methods of mathematics and metaphysics and his claim that mathematics admits of a greater degree of certainty. More generally, this illustrates the importance of Kant's reflections on mathematics for the (...) development of his Critical philosophy. (shrink)

This book is a welcome contribution to the literature on Kant's philosophy of mathematics in two particular respects. First, the author systematically traces the development of Kant's thought on mathematics from the very early pre-Critical writings through to the Critical philosophy. Secondly, it puts forward a challenge to contemporary Anglo-Saxon commentators on Kant's philosophy of mathematics which merits consideration.A central theme of the book is that an adequate understanding of Kant's pronouncements on mathematics must begin with the recognition that mathematics (...) in Kant's time was poised at the beginning of what Pierobon calls the ‘algebraic revolution’ of the nineteenth century. For Kant, Euclidean geometry, with its heavy reliance on the geometric image, was the paradigm of certainty. The algebraic revolution of the nineteenth century replaced that paradigm with an algebraic formalism, thereby freeing mathematics from any connection to the geometric image, and also severing the link to intuition. Pierobon describes this as the ‘divergence between the image and writing [l'écriture]’ . So great was the shift, Pierobon suggests, that, after the developments of the nineteenth century, it became difficult to find any sense in Kant's conception of mathematics as sensible knowledge. This, certainly, was the view of Russell, who notoriously claimed in Mysticism and Logic that modern developments in logic dealt a ‘fatal blow to the Kantian philosophy’ and that ‘the whole doctrine of a priori intuitions, by which Kant explained the possibility of pure mathematics, is wholly inapplicable to mathematics in its present form’.1 Pierobon claims, though, that much of Anglo-Saxon commentary on Kant's philosophy of mathematics begins from this ‘rationalist and logicist’ position , reading Kant's philosophy of mathematics from a post-algebraic-revolution perspective. This book attempts to offer a corrective to that position by offering a Kantian conception of mathematics …. (shrink)

There is a long tradition, in the history and philosophy of science, of studying Kant’s philosophy of mathematics, but recently philosophers have begun to examine the way in which Kant’s reflections on mathematics play a role in his philosophy more generally, and in its development. For example, in the Critique of Pure Reason , Kant outlines the method of philosophy in general by contrasting it with the method of mathematics; in the Critique of Practical Reason , Kant compares the Formula (...) of Universal Law, central to his theory of moral judgement, to a mathematical postulate; in the Critique of Judgement , where he considers aesthetic judgment, Kant distinguishes the mathematical sublime from the dynamical sublime. This last point rests on the distinction that shapes the Transcendental Analytic of Concepts at the heart of Kant’s Critical philosophy, that between the mathematical and the dynamical categories. These examples make it clear that Kant's transcendental philosophy is strongly influenced by the importance and special status of mathematics. The contributions to this book explore this theme of the centrality of mathematics to Kant’s philosophy as a whole. This book was originally published as a special issue of the Canadian Journal of Philosophy. (shrink)

There is a long tradition, in the history and philosophy of science, of studying Kant’s philosophy of mathematics, but recently philosophers have begun to examine the way in which Kant’s reflections on mathematics play a role in his philosophy more generally, and in its development. For example, in the Critique of Pure Reason , Kant outlines the method of philosophy in general by contrasting it with the method of mathematics; in the Critique of Practical Reason , Kant compares the Formula (...) of Universal Law, central to his theory of moral judgement, to a mathematical postulate; in the Critique of Judgement , where he considers aesthetic judgment, Kant distinguishes the mathematical sublime from the dynamical sublime. This last point rests on the distinction that shapes the Transcendental Analytic of Concepts at the heart of Kant’s Critical philosophy, that between the mathematical and the dynamical categories. These examples make it clear that Kant's transcendental philosophy is strongly influenced by the importance and special status of mathematics. The contributions to this book explore this theme of the centrality of mathematics to Kant’s philosophy as a whole. This book was originally published as a special issue of the Canadian Journal of Philosophy. (shrink)

This thesis attempts to argue against an influential interpretation of Kant's philosophy of mathematics according to which the role of pure intuition is primarily logical. Kant's appeal to pure intuition, and consequently his belief in the synthetic character of mathematics, is, on this view, a result of the limitations of the logical resources available in his time. In contrast to this, a reading is presented of the development of Kant's philosophy of mathematics which emphasises a much richer philosophical role for (...) intuition, beyond that of filling in deductive gaps in mathematical arguments. This role is largely determined by two factors: on the one hand, the epistemological gap in a traditional account of mathematical certainty, and on the other hand, the metaphysical worry raised by the followers of Leibniz and Wolff regarding the status of mathematics as a science consisting of truths. ;By appreciating how Kant's philosophy of mathematics developed against the background of what he saw as a conflict between the 'metaphysicians' and the 'mathematicians', we can see how the doctrine of pure intuition was required to fill the gaps in the account of the method and certainty of mathematics which Kant presents in the Prize Essay of 1764. This conflict takes its sharpest form between the monadists who uphold the metaphysical necessity of simple elements, and the geometers who uphold the infinite divisibility of space. In the Prize Essay Kant indirectly addresses this conflict by comparing the two disciplines. He asserts that the method of attaining certainty in mathematics is different from the method of metaphysics, and that as a result of this difference--primarily the different roles of definitions in each--mathematics is capable of a higher degree of certainty than metaphysics. He does not, however, explain this difference. Such an explanation seems particularly important in light of Kant's wish to secure geometry against the challenge of metaphysics. This thesis attempts to show that it is to provide such an explanation, to provide a philosophical grounding for the Prize Essay account, that Kant invokes the doctrine of pure intuition in his later account of mathematical knowledge. (shrink)